Question 78022: I need extra help with this problem how to work with it.
The length of a rectangle is 1cm more than 4 times its width. If the area of the rectangle is 74cm^2, find the dimensions of the rectangle to the nearest thousandth.
My responce to this is HUH...
Found 2 solutions by tutorcecilia, bucky:
Answer by tutorcecilia(2152)   (Show Source):
You can put this solution on YOUR website! Use the formula for the area of a rectangle:
A=(length)(width)
Identify the values:
A=74
Width = w =(unknown)
Length=4w+1
.
Plug all of the values into the formula and solve:
.
A=(length)(width)
74=(4w+1)(w)
74=4w^2+w
74-74=4w^2+w-74
0=4w^2+w-74
Use the quadratic formula to solve:
.
.
Two possible answers:
x=width=4.17797 or x=-4.4279 (this is an invalid answer because measurements are not negative)
.
So, the width = 4.17797 = 4.18
.
Length = 4w+1=4(4.17797)+1=17.7118 = 17.71
.
Check by plugging width=4.17797 and length=17.7118 back into the original equation and solve.
74=(4.17797)(17.7118)
74= 73.998 [close enough]
Answer by bucky(2189)  (Show Source):
You can put this solution on YOUR website! Since the problem involves the Area of a rectangle, let's start by recognizing that the
formula for the Area of a rectangle in terms of the length L and the width is:
.
A = L*W
.
and for this problem we are told that the Area is 74 cm^2. Substituting this value for A
in the equation for Area results in:
.
74 = L * W
.
We're done with that for the time being. Now we have to look at what other information
the problem tells us. The problem says that the Length is 1 cm more than 4 times the Width.
Four times the width is 4W. And 1 cm longer than 4W is 4W + 1. So wherever we see L we
can substitute the quantity 4W + 1. Let's go back to the Area equation:
.
74 = L * W
.
and in that equation let's substitute 4W + 1 for L. When we do, we get:
.
74 = (4W + 1) * W
.
This equation can now be solved because it contains only one unknown ... W. If it had
two unknowns, we would not be able to solve it directly. Anyhow, since it has only the one
unknown, let's solve it for W. Begin by multiplying out the right side of the equation to
get:
.
.
Now we need to get this into standard quadratic form ... everything on one side of the equal
sign and zero on the other side. We can do this by getting rid of the 74 on the left side.
Subtract 74 from the left side ... but if you do that, you must also subtract 74 from the
right side. This subtraction results in the equation becoming:
.
.
To get it into a little more conventional form, let's just transpose (or switch) sides to get:
.
.
The left side of this equation does not factor. Therefore, it probably is best to use the
quadratic formula to solve it. That formula tells you that for a quadratic of the form:
.
.
the possible solutions are given by:
.
.
Now by comparing our equation to the quadratic form we can see that W = x, 4 = a, 1 = b,
and -74 = c. By substituting these values into the equation for the possible solutions
we get two equations as follows:
.
 and
.
.
Let's simplify the top one of these two equations first. The -(1) becomes just -1, the
 is just 1, and the -4*4*(-74) equals +1184. In the denominator, the 2*4 equals 8.
Substituting these values results in:
.
.
Combining the two terms inside the radical sign results in 1185, and the square root of 1185
is (by calculator) 34.4238. Substituting this value results in the equation becoming:
.
.
So one value of W is 4.1780 cm. The second value of W is can be easily calculated.
Notice that it uses the same numbers as the first value, but there is a minus sign between
the two terms in the numerator. So it becomes:
.
.
We can stop right here because we can see that the numerator will be negative, making the
value of W negative ... and what sense does it make to say a rectangle with a negative
width? None whatsoever. So we ignore this negative answer. We then know that the only
acceptable value for W is 4.1780 cm.
.
Now go back to our original equation for L. We said that L was equal to 4W + 1. And now we
know that W is 4.1780 cm. So we can write that:
.
L = (4*4.1780) + 1 = 16.712 + 1 = 17.712 cm
.
So our two answers are: the width = 4.1780 cm and the length = 17.712 cm. Rounding
these to the nearest hundredth as the problem asks you to do, results in:
.
Width = 4.18 cm and Length = 17.71 cm
.
Now, as a check, you can multiply the length times the width and you find that this results
in an area of 74.0278 sq cm ... pretty close to the given 74 sq cm given by the problem.
A difference this small results from rounding off the length and width to the nearest
hundredth.
.
Hope this helps you to get from HUH to "got it".
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